Energy conserving local discontinuous Galerkin methods for wave propagation problems Yulong Xing1 , Ching-Shan Chou2 and Chi-Wang Shu3 Abstract Wave propagation problems arise in a wide range of applications. The energy conserving property is one of the guiding principles for numerical algorithms, in order to minimize the phase or shape errors after long time integration. In this paper, we develop and analyze a local discontinuous Galerkin (LDG) method for solving the wave equation. We prove optimal error estimates, superconvergence toward a particular projection of the exact solution, and the energy conserving property for the semi-discrete formulation. The analysis is extended to the fully discrete LDG scheme, with the centered second-order time discretization (the leap-frog scheme). Our numerical experiments demonstrate optimal rates of convergence and superconvergence. We also show that the shape of the solution, after long time integration, is well preserved due to the energy conserving property.
Keywords: wave propagation, local discontinuous Galerkin method, energy conservation, error estimate, superconvergence 1
Computer Science and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831 and Department of Mathematics, University of Tennessee, Knoxville, TN 37996. E-mail:
[email protected]. Fax: (865) 574-0680. Research is sponsored by the Office of Advanced Scientific Computing Research; U.S. Department of Energy. The work was performed at the ORNL, which is managed by UT-Battelle, LLC under Contract No. DE-AC05-00OR22725. 2 Department of Mathematics, The Ohio State University, Columbus, OH 43221. Email:
[email protected]. Research is supported by NSF grant DMS-1020625. 3 Division of Applied Mathematics, Brown University, Providence, RI 02912. E-mail:
[email protected]. Research is supported by DOE grant DE-FG02-08ER25863 and NSF grant DMS-1112700.
1
1
Introduction
Wave propagation problems arise in science, engineering and industry, and they are significant to geoscience, petroleum engineering, telecommunication, and the defense industry (see [13, 20] and the references therein). Among all the equations describing wave propagation, the constant coefficient wave equation is the simplest one and has been extensively studied. One of the most important properties of the wave equation is the conservation of energy. Experiences reveal that energy conserving numerical methods, which conserve the discrete approximation of energy, are favorable because they are able to maintain the phase and shape of the waves accurately. Numerical methods without this property may result in substantial phase and shape errors after long time integration. A vast amount of literature can be found on the numerical approximation of the wave equation. All types of numerical methods, including finite difference, finite element, finite volume and spectral methods have their proponents. Here, we will confine our attention in finite element methods, in particular, discontinuous Galerkin (DG) methods. The DG methods belong to a class of finite element methods using discontinuous piecewise polynomial spaces for both the numerical solution and the test functions. They were originally devised to solve hyperbolic conservation laws with only first order spatial derivatives, e.g. [7, 8, 9, 11, 12]. They allow arbitrarily unstructured meshes, and have compact stencils. Moreover, they easily accommodate arbitrary h-p adaptivity. The DG methods were later generalized to the local discontinuous Galerkin (LDG) methods by Cockburn and Shu to solve convectiondiffusion equations [10], motivated by successful numerical experiments from Bassi and Rebay [2] for the compressible Navier-Stokes equations. Methods for solving the wave equation can be divided into two categories. The first category is to rewrite the wave equation into a first order hyperbolic system. Therefore, various DG methods designed for first order hyperbolic systems can be applied, for example the standard DG method proposed by Cockburn and Shu [9]. High order nodal DG methods for Maxwell’s equations of a first order system are proposed in [19]. Space-time DG methods 2
are studied by Falk and Richter in [14], and later by Monk and Richter in [21]. A suboptimal DG method using central fluxes, which is also energy conserving, has been presented in [15] for Maxwell’s equations. More recently, Chung and Engquist [4, 5] have proposed an optimal, energy conserving DG method for the wave equation using staggered grids. The second category of numerical methods is to tackle the second derivatives in the wave equation directly, without introducing more unknowns to reduce the order of the system. A non-symmetric interior penalty discontinuous Galerkin (IPDG) method has been presented by Rivi`ere and Wheeler in [22]. Later Grote et al. [17, 18] proposed a symmetric IPDG method for the second order equation, which conserves a specifically defined energy. Adjerid and Temimi [1] presented a Galerkin method which uses finite element spaces of continuous functions in space, and discontinuous functions in time, and their method achieves the optimal convergence rate and superconvergence property. In this paper, we primarily consider the one-dimensional linear wave equation utt = uxx ,
in [a, b] × [0, T ],
(1.1)
subject to the initial conditions u(x, 0) = u0 (x),
ut (x, 0) = v0 (x),
(1.2)
and a periodic boundary condition u(a, t) = u(b, t),
t ≥ 0.
DG method for this problem with the upwind flux has optimal high order accuracy, but is dissipative. On the other hand, DG method with central flux is energy conserving, but is only suboptimal accurate. Usually it is difficult to obtain DG schemes for wave equations which are non-dissipative (energy conserving for the physical energy) and optimal high order accurate, without going to staggered mesh [4, 5] or using penalty with a parameter to adjust [17, 18]. Here, we consider the local discontinuous Galerkin method defined on a single mesh for this linear wave problem. The proposed semi-discrete scheme is shown to be 3
energy conserving. It has the optimal convergence rates in both the energy and L2 norms, and the upper bound of the errors grows in time only in a linear fashion. This method can achieve superconvergence, towards a particular projection of the exact solution. The order of superconvergence is proved to be k + 3/2 when piecewise P k polynomials are used. Coupled with the centered second-order time discretization (the leap-frog method) for the temporal derivatives, we present the fully discrete LDG method, which is explicit in time, and we also show the energy conservation property. To demonstrate the importance of energy conservation, we test an example with long time integration, and the numerical results reveal that our method stays very accurate after long time integration, in contrast to numerical methods without this property. This paper is organized as follows. In Section 2, we present the semi-discrete LDG method, and prove its energy conserving property. The optimal error estimates, both in the energy norm and the L2 norm, are analyzed in Section 3, and therein, the upper bound of errors is proved to grow linearly in time. In Section 4, we prove the superconvergence of the LDG method toward a particular projection of the exact solution. The fully discrete LDG method, with the leap-frog time discretization, and its energy conserving properties are presented in Section 5. Section 6 contains numerical experiments that demonstrate the optimal convergence rates and energy conservation of the proposed LDG method, and a comparison with the IPDG method proposed in [17]. Finally, we give the concluding remarks in Section 7.
2 2.1
Local discontinuous Galerkin discretization Notations
We divide the interval I = [a, b] into N subintervals and denote the cells by Ij = [xj− 1 , xj+ 1 ] 2
for j = 1, · · · , N. The center of each cell is xj =
1 (xj− 1 2 2
2
+ xj+ 1 ), and the mesh size is 2
denoted by hj = xj+ 1 − xj− 1 , with h = max1≤j≤N hj being the maximal mesh size. We 2
2
assume that the mesh is regular, namely, the ratio between the maximal and the minimal 4
mesh sizes stays bounded during mesh refinement. The piecewise polynomial space Vhk is defined as the space of polynomials of degree up to k in each cell Ij , that is, Vhk = {v : v|Ij ∈ P k (Ij ), j = 1, 2, · · · , N}.
(2.1)
Note that functions in Vhk are allowed to have discontinuities across element interfaces. The solution of the numerical scheme is denoted by uh , which belongs to the finite element and (uh )− the limit values of uh at xj+ 1 from the right space Vhk . We denote by (uh )+ j+ 1 j+ 1 2
2
2
− cell Ij+1 and from the left cell Ij , respectively. We use the usual notations [uh ] = u+ h − uh − and u¯h = 12 (u+ h + uh ) to respectively represent the jump and the mean of the function uh at
the element interfaces. The L2 norm over the interval I is denoted by k · k.
2.2
The LDG method
In this subsection, we define the semi-discrete LDG method for the wave equation (1.1), by discretizing the space with the LDG method and leaving the time dependence continuous. First, we write the wave equation into a first order system by substituting ux with variable q: utt = qx ,
(2.2)
q = ux . The LDG method for (2.2) is then formulated as follows: find uh , qh ∈ Vhk , such that Z
Z
qh v + )j− 1 = 0, (uh )tt vdx + qh vx dx − (ˆ qh v − )j+ 1 + (ˆ 2 2 I I Zj Z j uh w + )j− 1 = 0, qh wdx + uh wx dx − (ˆ uh w − )j+ 1 + (ˆ Ij
2
Ij
2
(2.3) (2.4)
for all test functions v, w ∈ Vhk . The hatted terms, qˆh and uˆh , in (2.3)-(2.4) are the cell boundary terms obtained from integration by parts, and they are the so-called numerical fluxes. These numerical fluxes are single-valued functions defined on the cell boundaries and should be designed according to guiding principles for different PDEs to ensure numerical 5
stability. Here we use the simple alternating fluxes: uˆh = u+ h,
qˆh = qh− ,
(2.5)
in which we have omitted the half-integer indices j + 21 , as all quantities in (2.5) are computed at the same points (i.e., the cell interface). We remark that the choice of the fluxes (2.5) is not unique. We can, for example, alternatively choose the numerical fluxes to be qˆh = qh+ ,
2.3
uˆh = u− h.
(2.6)
Energy conservation
As is well-known, the one-dimensional linear wave equation (1.1) admits an important conR served quantity – the energy, E = I (u2t + u2x )dx. Experiences show that schemes conserving the discrete analogs of energy often produce approximations that behave better for long time
simulation. In this subsection, we will show that the proposed semi-discrete LDG method conserves energy. Proposition 2.1. The (continuous) energy Z Eh (t) = (uh )2t + qh2 dx
(2.7)
I
is conserved by the the semi-discrete LDG method (2.3)-(2.4) for all time.
Proof. We first take the time derivative of Eq. (2.4) and choose the test function w = qh to obtain Z
(qh )t qh dx + Ij
Z
Ij
− + + (uh )t (qh )x dx − ((uh )+ t qh )j+ 1 + ((uh )t qh )j− 1 = 0. 2
2
In Eq. (2.3), we choose the test function to be (uh )t : Z Z − + (uh )tt (uh )t dx + qh (uh )tx dx − (qh− (uh )− t )j+ 1 + (qh (uh )t )j− 1 = 0. Ij
2
Ij
2
Adding Eq. (2.8) to Eq. (2.9), one obtains Z Z − + (uh )tt (uh )t dx + qh (uh )tx dx − (qh− (uh )− t )j+ 21 + (qh (uh )t )j− 21 Ij I Z Z j − + + + (qh )t qh dx + (uh )t (qh )x dx − ((uh )+ t qh )j+ 1 + ((uh )t qh )j− 1 = 0. Ij
2
Ij
6
2
(2.8)
(2.9)
R Using integration by parts on the term Ij qh (uh )tx dx, we get Z Z + − (uh )tt (uh )t dx + (qh )t qh dx + (qh− (uh )+ t )j− 1 − ((uh )t qh )j+ 1 = 0. Ij
2
Ij
2
(2.10)
By summing up Eq. (2.10) over all cells and using the periodic boundary conditions, we have d dt
Z
I
((uh )2t
+
qh2 )dx
= 0.
Therefore the quantity Eh is invariant in time.
3
Error estimate
In this section, we derive the optimal error estimates for the energy conserving LDG method (2.3)-(2.4) proposed in Section 2. The error estimate in the energy norm will be carried out first, and then the analysis will be extended to the L2 norm. We will also show that these error bounds are both linear in time. We start by introducing the projections and other notations to be used throughout this paper. The standard L2 projection of a function ω(x) with k + 1 continuous derivatives into space Vhk is denoted by Ph , i.e., Z
Ph ωvdx =
Ij
Z
ωvdx,
Ij
for any v ∈ P k on Ij . In addition, we define Ph− ω to be a projection of ω into Vhk , such that Z Z − Ph ωvdx = ωvdx, Ij
Ij
for any v ∈ P k−1 on Ij , and (Ph− ω)− = ω −
at xj+ 1 . 2
Similarly, the projection Ph+ ω is defined as the projection of ω into Vhk , such that Z Z + Ph ωvdx = ωvdx, Ij
Ij
for any v ∈ P k−1 on Ij , and (Ph+ ω)+ = ω + 7
at xj− 1 . 2
For these projections, it is easy to show (see [6]): 1
kω e k + hkω e k∞ + h 2 kω e kΓh ≤ Chk+1 ,
(3.1)
where ω e = ω − Ph ω or ω e = ω − Ph± ω, and Γh denotes the set of boundary points of all cells. The constant C depends on the function ω, but is independent of the mesh size h. Let us denote the errors by eu = u − uh ,
εu = u − Ph+ u,
e¯u = Ph+ u − uh ,
eq = q − qh ,
εq = q − Ph− q,
e¯q = Ph− q − qh ,
(3.2)
which, from left to right, respectively represent the errors between the exact solution and the numerical solution, the projection errors, and the errors between the numerical solution and the particular projection of the exact solution. Note that the signs of the projection Ph± of u and q in (3.2) are consistent with the choice of the numerical fluxes in (2.5). So if the other set of numerical fluxes (2.6) are chosen, the signs of Ph± in (3.2) should be changed accordingly. To obtain the error estimates and the superconvergence property of the proposed LDG method, the projections of the initial conditions for the numerical scheme need to be carefully chosen. Note that we have two initial conditions in (1.2), one for u and the other for ut . We take the initial condition uh (x, 0) as Ph+ u(x, 0) = Ph+ u0 (x), which is consistent with the choice of the numerical fluxes (2.5). The other initial condition (uh )t (x, 0) is given by the standard L2 projection. Thus, we have the following lemma. Lemma 3.1. Assume the initial conditions of the LDG scheme (2.3)-(2.4) are given by uh (x, 0) = Ph+ u(x, 0),
(uh )t (x, 0) = Ph ut (x, 0),
(3.3)
there holds the following error estimate k¯ eu (0)k = 0,
k¯ eq (0)k ≤ Chk+1,
8
k(¯ eu )t (0)k ≤ Chk+1 ,
(3.4)
and Z
(eu )t (0)vdx = 0,
for any v ∈ P k .
(3.5)
Ij
Proof. Here we only give the proof for the error estimate of k¯ eq (0)k. All the other results can be obtained similarly. Subtracting (2.4) of the LDG method from the weak formulation satisfied by the exact solution q yields R
e wdx + Ij q
R
Ij
+ + − eu wx dx − (e+ u w )j+ 1 + (eu w )j− 1 = 0, 2
(3.6)
2
for any test function w ∈ Vhk . By the properties of the projection Ph+ , we can derive Z
eq wdx +
Ij
Z
Ij
+ − e+ e¯u wx dx − (¯ e+ u w )j− 1 = 0. u w )j+ 1 + (¯ 2
(3.7)
2
Since e¯u (0) = uh (0) − Ph+ u(0) = 0, we have Z
eq (0)wdx = 0,
Ij
for any test function w ∈ Vhk . Taking w = e¯q (0) yields 1 k¯ eq (0)k = (¯ eq (0)) dx = − e¯q (0)εq (0)dx ≤ 2 I I 2
Z
2
Z
Z
1 (¯ eq (0)) dx + 2 I 2
Z
(εq (0))2 dx, I
and therefore, k¯ eq (0)k ≤
Z
12 = O(hk+1 ). (εq (0)) dx 2
I
(3.8)
This completes the proof. Based on initial conditions (3.3), we have the following error estimate in the energy norm. Proposition 3.1. Let u and q be the exact solutions of the wave equation (2.2), and uh , qh be the numerical solutions of the semi-discrete LDG method (2.3)-(2.4), with the numerical fluxes defined in (2.5) and the initial conditions (3.3). Consider a regular discretization of
9
I and the piecewise polynomial finite element space Vhk , for small enough h, there holds the following error estimates: k(eu )t k ≤ Chk+1 (t + 1),
keq k ≤ Chk+1 (t + 1),
(3.9)
where the constant C depends on kukk+3 and kut kk+2. Proof. By subtracting the LDG method (2.3)-(2.4) from the weak formulation satisfied by the exact solutions u and q, we can derive the error equations Z Z − + − (eu )tt vdx + eq vx dx − (e− q v )j+ 21 + (eq v )j− 21 = 0, I I Zj Z j + + − eq wdx + eu wx dx − (e+ u w )j+ 1 + (eu w )j− 1 = 0, Ij
2
Ij
2
(3.10) (3.11)
for all test functions v, w ∈ Vhk . Using the properties of the projections Ph± , the error equations are equivalent to Z Z − + (eu )tt vdx + e¯q vx dx − (¯ e− e− q v )j+ 21 + (¯ q v )j− 21 = 0, I I Zj Z j + − e+ eq wdx + e¯u wx dx − (¯ e+ u w )j− 1 = 0, u w )j+ 1 + (¯ Ij
2
Ij
2
(3.12) (3.13)
Along the same line in the proof of Proposition 2.1, we first take the time derivative of Eq. (3.13), choose the test functions w = e¯q , v = (¯ eu )t , and sum up the resulting two equations to get Z
(eu )tt (¯ eu )t dx + Ij
Z
Ij
(eq )t e¯q dx + (¯ e− eu )+ e− eu )+ q (¯ t )j− 1 − (¯ q (¯ t )j+ 1 = 0. 2
2
By summing up the above equation over all cells and using the periodic boundary conditions, we have Z
(eu )tt (¯ eu )t dx +
I
Z
(eq )t e¯q dx = 0
I
Therefore, Z 1d 1d 2 2 k(¯ eu )t k + k¯ eq k = (¯ eu )2t + e¯2q dx 2 dt 2 dt I Z Z ≤ (εu )tt (¯ eu )t dx + (εq )t e¯q dx ≤ k(εu )tt kk(¯ eu )t k + k(εq )t kk¯ eq k I
I
k+1
≤ C1 h
1
k(¯ eu )t k + C2 hk+1 k¯ eq k ≤ Chk+1 (k(¯ eu )t k2 + k¯ eq k2 ) 2 , 10
which leads to 1 d k(¯ eu )t k2 + k¯ eq k2 2 ≤ Chk+1 . dt Combining this inequality with the property of the initial condition (3.4), we conclude that k(¯ eu )t k2 + k¯ eq k2
12
≤ C(t + 1)hk+1,
in which the constant C only depends on kukk+3 and kut kk+2 . Together with the optimal projection error (3.1), the error estimate (3.9) follows.
Next, we consider the error estimate with respect to the L2 norm. Proposition 3.2. Let u and q be the exact solutions of the wave equation (2.2), and uh , qh be the numerical solutions of the semi-discrete LDG method (2.3)-(2.4), with the numerical fluxes defined in (2.5) and the initial conditions (3.3). Consider a regular discretization of I and the piecewise polynomial finite element space Vhk , for small enough h, there holds the following error estimates: max keu (t)k ≤ Chk+1 (T + 1),
(3.14)
t∈[0,T ]
where the constant C only depends on the solution u. Proof. We split (eu )tt as the summation of (eu )tt and (εu )tt in Eq. (3.12), and use chain rule in time derivative to obtain −
Z
(¯ eu )t vt dx+
Ij
Z
Ij
+ − e− e¯q vx dx−(¯ e− q v )j− 21 q v )j+ 21 +(¯
d = − (εu )tt vdx− dt Ij Z
Z
(¯ eu )t vdx. (3.15)
Ij
For any time τ ≤ T , we denote the time integral of the error by E¯u (t) =
Z
τ
e¯u (s)ds,
¯q (t) = E
t
Z
τ
e¯q (s)ds,
Eq (t) =
t
Z
τ
Eqε (t)
eq (s)ds,
t
=
Z
τ
εq (s)ds.
t
Taking the integral of Eq. (3.13) in time, from t to τ , gives Z
Ij
Eq wdx +
Z
Ij
¯ + w + ) 1 = 0. ¯u wx dx − (E¯ + w − ) 1 + (E E u j− u j+ 2
11
2
(3.16)
If we choose the test functions to be v = E¯u (t) and w = e¯q (t) in (3.15)-(3.16), and use the fact that vt = −¯ eu (t), we have Z Z Z Z d − + − − (¯ eu )t E¯u dx, eq E¯u )j− 1 = − (εu )tt E¯u dx − (¯ eu )t e¯u dx + e¯q (E¯u )x dx − (¯ eq E¯u )j+ 1 + (¯ 2 2 dt Ij Ij Ij Ij Z Z ¯ + ¯+ ) 1 = 0. Eq e¯q dx + E¯u (¯ eq )x dx − (E¯u+ e¯− q )j+ 1 + (Eu e q j− Ij
2
Ij
2
Adding up these two equations and summing over all cells, one obtains Z Z Z 1d d 2 2 ¯ ¯ ¯ (¯ eu )t Eu dx − Eqε e¯q dx k¯ eu k − kEq k = − (εu )tt Eu dx − 2 dt dt ZI Z I Z I d (eu )t E¯u dx − Eqε e¯q dx. = − (εu )t e¯u dx − dt I I I By integrating the inequality from 0 to τ , we get 1 1 1 ¯ 2 k¯ eu (τ )k2 − k¯ eu (0)k2 + kE q (0)k 2 2 2 Z τZ Z Z ¯ =− (εu )t e¯u dxdt + (eu )t (0)Eu (0)dx − 0
I
I
0
τ
Z
I
Eqε e¯q dxdt,
(3.17)
¯q (τ ) = E¯u (τ ) = 0 is used. By the properties of the projection (3.1), we in which the fact E have k(εu )t k = Chk+1 . Note that Z τ Z τ Z ε − Eq (t) = εq (s)ds = q(s) − Ph q(s) ds = t
t
τ
q(s)ds −
t
Ph−
Z
τ
q(s)ds ,
t
and therefore we can conclude that kEqε k = Chk+1 . Combining with the property of the L2
projection (3.5), we have 1 1 ¯ 2 k¯ eu (0)k2 + kE q (0)k 2 Z Z 2 Z τ Z τ ε ≤ (εu )t e¯u dxdt + Eq e¯q dxdt Z 0τ I Z τ0 I ≤ k(εu )t kk¯ eu kdt + kEqε kk¯ eq kdt
1 k¯ eu k2 (τ ) − 2
0
0
≤ τ max k(εu )t k max k¯ eu k + τ max kEqε k max k¯ eq k t∈[0,τ ] t∈[0,τ ] t∈[0,τ ] t∈[0,τ ] ε ≤ T max k(εu )t k max k¯ eu k + max kEq k max k¯ eq k t∈[0,T ] t∈[0,T ] t∈[0,T ] t∈[0,T ] k+1 2k+2 ≤ CT h max k¯ eu k + (T + 1)h t∈[0,T ]
≤ C(T 2 + 1)h2k+2 +
1 max k¯ eu k2 . 4 t∈[0,T ]
12
Since this is true for any τ < T , we have 1 1 1 1 max k¯ eu k2 − k¯ eu (0)k2 + kE¯q (0)k2 ≤ C(T 2 + 1)h2k+2 + max k¯ eu k2 . 2 t∈[0,T ] 2 2 4 t∈[0,T ] Hence, 1 1 1 max k¯ eu k2 + kE¯q (0)k2 ≤ C(T 2 + 1)h2k+2 + k¯ eu (0)k2 = C(T 2 + 1)h2k+2 , 4 t∈[0,T ] 2 2 from which we can conclude max keu (t)k ≤ Chk+1 (T + 1).
t∈[0,T ]
where the constant C only depends on the solution u.
4
Superconvergence
The superconvergence property of the LDG method is studied in this section. We will prove superconvergence towards a particular projection of the exact solution, with the order k+3/2, in which an increase of 1/2 over the optimal error estimate of Section 3 can be observed. To obtain the superconvergence property of the method, two functionals related to the L2 norm of a function f (x) on Ij , as defined in [3], are needed: Z x − xj−1/2 d x − xj − f (x) dx, Bj (f ) = f (x) hj dx hj Ij Z x − xj+1/2 d x − xj + f (x) dx. Bj (f ) = f (x) hj dx hj Ij Properties of these functionals in the following lemma are essential to the proof of the superconvergence. The proof can be found in [3] and is therefore omitted. Lemma 4.1. For any function f (x) ∈ C 1 on Ij , we have Z f 2 (xj+1/2 ) 1 − f 2 (x)dx + , Bj (f ) = 4hj Ij 4 Z f 2 (xj−1/2 ) 1 + f 2 (x)dx − . Bj (f ) = − 4hj Ij 4 (See [3] for the detailed proof ) 13
(4.1) (4.2)
We prove the superconvergence of the numerical solution in the following Proposition. Proposition 4.1. Let u and q be the exact solutions of the wave equation (2.2), and uh , qh be the numerical solutions of the semi-discrete LDG method (2.3)-(2.4), with the numerical fluxes defined in (2.5) and the initial conditions (3.3). Consider a regular discretization of I and the piecewise polynomial finite element space Vhk , for small enough h, there holds the following error estimate: 3
k¯ eu k ≤ Chk+ 2 (t + 1),
(4.3)
where the constant C only depends on the solution u. Proof. In the proof of Proposition 3.2, we have derived an upper bound for the error term k¯ eu (τ )k2 in Eq. (3.17). Utilizing the property (3.5) due to the L2 projection of the initial condition (uh )t (0), Eq. (3.17) becomes 1 1 ¯ 1 2 k¯ eu (τ )k2 − k¯ eu (0)k2 + kE q (0)k = − 2 2 2
τ
Z
Z
0
(εu )t e¯u dxdt − I
Z
0
τ
Z
I
Eqε e¯q dxdt,
(4.4)
for any time τ ≤ T . Firstly, we consider the first term on the right-hand side and tackle the term e¯u . We rewrite the error equation (3.13) as Z
eq wdx − Ij
Z
Ij
(¯ eu )x wdx − [¯ eu ]w + j− 1 = 0,
(4.5)
2
by performing integration by parts on the term
R
Ij
e¯u wx dx. Define e¯u = rj + dj (x)(x − xj )/hj
on the cell Ij , where rj = e¯u (xj ) is a constant and dj (x) ∈ P k−1 . By choosing the test + function w in (4.5) to be dj (x)(x − xj− 1 )/hj on Ij , the last term wj− 1 will be reduced to 0. 2
Using the definition of
Bj− (f ), Z
Ij
2
one has
eq dj (x)(x − xj− 1 )/hj dx − Bj− (dj (x)) = 0. 2
By Lemma 4.1, this is equivalent to Z
eq dj (x) Ij
x − xj− 1
2
hj
1 dx − 4hj
Z
14
Ij
d2j (x)dx −
d2j (xj+1/2 ) = 0, 4
and hence, Z
Ij
d2j (x)dx
≤4
Z
Ij
eq dj (x)(x − xj− 1 )dx.
(4.6)
2
We define piecewise polynomials d(x) and φ1 (x), such that d(x) = dj (x), and φ1 (x) = x−xj− 1 2
on Ij . Clearly, kφ1 k∞ = maxj hj = h, and Eq. (4.6) leads to kdk2 ≤ 4keq kkdkkφ1k∞ ≤ 4hkeq kkdk. Therefore, by Proposition 3.1, we have kdk ≤ 4hkeq k ≤ Chk+2 (t + 1). Secondly, we repeat similar analysis to the term E¯q in (4.4). From Eq. (3.12), we can derive Z
(eu )tt vdx −
Ij
Z
Ij
(¯ eq )x vdx − [¯ eq ]v − j+ 1 = 0. 2
Integrating the above equation in time from t to τ gives Z Z ((eu )t (τ ) − (eu )t (t))vdx − (E¯q )x vdx − [E¯q ]v − j+ 1 = 0. Ij
Ij
(4.7)
2
Similar to the previous analysis, we define E¯q = bj + sj (x)(x − xj )/hj on the cell Ij , where bj is a constant and sj (x) ∈ P k−1. Choose the test function v in (4.7) to be sj (x)(x − xj+ 1 )/hj 2
on Ij , and use the definition of Bj+ (f ) and Lemma 4.1 to obtain Z Z x − xj+ 1 s2j (xj−1/2 ) 1 2 ((eu )t (τ ) − (eu )t (t)) sj (x) s2j (x)dx + dx + = 0. hj 4hj Ij 4 Ij Hence, Z s2j (x)dx ≤ 4 ((eu )t (τ ) − (eu )t (t)) sj (x)(x − xj+ 1 )dx . 2 Ij Ij
Z
(4.8)
Then we define piecewise polynomials s(x) and φ2 (x), such that s(x) = sj (x), and φ2 (x) = x − xj+ 1 on Ij , and finally get 2
ksk2 ≤ 4k(eu )t (τ ) − (eu )t (t)kkskkφ2k∞ ≤ 4hk(eu )t (τ ) − (eu )t (t)kksk. 15
By Proposition 3.1, we conclude that ksk ≤ 4k(eu )t (τ ) − (eu )t (t)k ≤ Chk+2 (τ + 1). Lastly, we will use the above results to bound the right-hand side of Eq. (4.4). By the definition of projections, (εu )t and Eqε are both orthogonal to piecewise constant function, hence Z
(εu )t e¯u dx =
I
XZ j
=
XZ j
Z
I
Eqε e¯q dx
= =
Ij
XZ j
(εu )t dj (x)(x − xj )/hj dx,
Ij
XZ j
(εu )t (rj + dj (x)(x − xj )/hj )dx
Ij
Ij
Eqε (bj + sj (x)(x − xj )/hj )dx Eqε sj (x)(x − xj )/hj dx.
Define the function φ3 (x) = (x − xj )/hj on Ij , then kφ3 k∞ = 12 . Therefore, 1 1 1 k¯ eu (τ )k2 − k¯ eu (0)k2 + kE¯q (0)k2 2 2 Z 2 τ
k(εu )t kkφ3 k∞ kdk + kEqε kkφ3 k∞ ksk dt Z0 τ k+1 1 k+2 k+1 1 k+2 ≤ Ch Ch (t + 1) + Ch Ch (τ + 1) dt 2 2 0 Z τ = C(t + τ + 1)h2k+3 dt = C(τ + 1)2 h2k+3 .
≤
0
Combining this inequality with the initial condition (3.4), and replacing τ by t, we have
Z t
2
1 2 2 2k+3 k¯ eu (t)k + , e¯q (s)ds
≤ C(t + 1) h 2 0
and in particular
3
k¯ eu (t)k ≤ C(t + 1)hk+ 2 , where the constant C only depends on the solution u. Although the proofs for both the error estimate in Section 3 and the superconvergence in Section 4 are shown with the numerical fluxes (2.5), the same results hold for the other choice of fluxes (2.6). 16
5
Time discretization
In order to extend the energy conservation property of the the semi-discrete method to the fully discrete method, it is natural to employ time stepping methods conserving discrete energy. In this paper, we consider the second order leap-frog method for time discretization, which is well-known to be energy conserving. Let 0 ≤ t0 < t1 < · · · < tN = T be a partition of the interval [0, T ] with time step ∆tn = tn+1 − tn . Here uniform time step ∆t is used. The fully discrete approximations unh to u(·, tn ) are constructed as follows: for n = 1, . . . , N − 1, un+1 ∈ Vhk is given by h Z un+1 − 2unh + uhn−1 h qh v + )j− 1 = 0, vdx + qhn vx dx − (ˆ qh v − )j+ 1 + (ˆ 2 2 2 ∆t Ij I Zj Z uh w + )j− 1 = 0, qhn wdx + unh wx dx − (ˆ uh w − )j+ 1 + (ˆ Z
Ij
2
Ij
2
(5.1) (5.2)
for all test functions v and w in Vhk , and the numerical fluxes uˆh , qˆh are defined as in (2.5) or (2.6). In Proposition 2.1, we showed the semi-discrete LDG method conserves the continuous energy Eh (t). Along the same line of analysis, we can exhibit the following conserved quantity for the fully discrete method. Proposition 5.1. The solution to the fully discrete leap-frog LDG method (5.1)-(5.2), conserves the (discrete) energy Ehn+1 for all n.
n+1
uh − unh 2 qhn+1 + qhn 2 ∆t2 qhn+1 − qhn 2
=
+
− 4
∆t 2 ∆t
(5.3)
Proof. Suppose the numerical fluxes (2.5) are used, as the following proof can be carried out n+1 n n−1 un un+1 −un−1 1 uh −uh h −uh h h + = similarly for fluxes (2.6). Choose the test function v = 2 ∆t ∆t 2∆t
17
in (5.1) to obtain Z
Ij
n+1 Z uh − uhn−1 − uhn−1 un+1 − 2unh + uhn−1 un+1 n h h dx dx + qh ∆t2 2∆t 2∆t Ij x ! ! n+1 n+1 n−1 − n−1 + u u − u − u h h h h − (qhn )− + (qhn )− 2∆t 2∆t 1 j+ 2
(5.4) = 0. j− 21
Consider Eq. (5.2) at time levels tn−1 and tn+1 , and let the test function w be
1 n q . 2∆t h
Sub-
tracting these two equations yields Z
Ij
Z un+1 − uhn−1 n qhn+1 − qhn−1 n h qh dx + (qh )x dx (5.5) 2∆t 2∆t Ij n+1 + n+1 + (uh ) − (uhn−1 )+ n − (uh ) − (uhn−1 )+ n + (qh ) (qh ) − + = 0. 2∆t 2∆t j+ 1 j+ 1 2
2
Adding Eq. (5.4) to (5.5) and summing over all cells gives Z n+1 qh − qhn−1 n un+1 − 2unh + uhn−1 un+1 − uhn−1 h h dx + qh dx ∆t2 2∆t 2∆t Ij I Z n+1 Z n+1 qh + 2qhn + qhn−1 qhn+1 − qhn−1 n − uhn−1 uh − 2unh + uhn−1 un+1 h dx + qh dx = ∆t2 2∆t 4 2∆t Ij I Z n+1 qh − 2qhn + qhn−1 qhn+1 − qhn−1 n qh dx − 4 2∆t Ij
n+1
uh − unh 2 qhn+1 + qhn 2 ∆t2 qhn+1 − qhn 2 1
+
−
=
2∆t ∆t 2 4 ∆t
n
!
uh − uhn−1 2 qhn + qhn−1 2 ∆t2 qhn − qhn−1 2
−
+
−
∆t 2 4 ∆t
Z
= 0.
Therefore, by the definition of Ehn in (5.3), we have Ehn+1 = Ehn for all n, which completes the proof. Remark 5.1. The discrete energy Ehn can be rewritten as: Ehn+1
=
Z I
un+1 − unh h ∆t
2
dx +
Z
I
qhn+1qhn dx,
which is a consistent approximation of the continuous energy (2.7). 18
(5.6)
6
Numerical experiments
Numerical experiments designed to gauge the performance of our conservative schemes are reported in this section. Attention is given particularly to two issues: 1. Validation of the theoretical results, including a study of the convergence and superconvergence rates. 2. Investigating the long time behavior of our energy conserving scheme, which is also compared with the performance of the energy conserving IPDG methods proposed in [18]. This includes a comparison of the errors as a function of time, and the solutions after long time integration.
6.1
Convergence rates
In this subsection, we test the order of convergence and superconvergence of the proposed LDG scheme. The results here present the case of uniform meshes, in which the domain is uniformly divided into N cells. Since the second order leap-frog time discretization is employed and our interest is in the effect of the spatial discretization, we determine the timestep by the relation ∆t = Ch2 . This relation guarantees that the error will be dominated by the spatial discretization. We consider the wave equation utt = uxx ,
x ∈ [0, 1]
with initial conditions u(x, 0) = sin(πx),
ut (x, 0) = −π cos(πx),
and a periodic boundary condition u(0, t) = u(1, t) for all t ≥ 0. This problem was considered in [16] and has the exact solution u(x, t) = sin(πx) cos(πt). 19
We implemented the LDG method with the alternating fluxes (2.5) and take ∆t = 0.01h2 . Since leap-frog method requires initial conditions for two time steps, we consider Taylor expansion of u at t = 0: u(x, ∆t) = u(x, 0) + ∆tut (x, 0) +
∆t3 ∆t2 utt (x, 0) + uttt (x, 0) + O(∆t4 ), 2 6
and convert the higher derivatives of t to derivatives of x by repeatedly using the wave equation, while u and ut are given by initial conditions. To obtain the desired order of convergence of u, following the initial conditions (3.3), we take the projection Ph− of u(x, 0), and L2 projection Ph of ut (x, 0); that is, with u(x, 0) denoted by u0 and ut (x, 0) by v0 , to use the initial conditions: u0h = Ph− u0 , u1h = u0h + ∆tPh v0 +
∆t3 ∆t2 0 (uh )xx + (Ph v0 )xx . 2 6
Tables 6.1 - 6.3 list the numerical errors and the orders of convergence for P k spaces, k = 1, 2, 3. In each table, the L2 -norm of the errors eu , e¯u , and eq at final time T = 1 are presented. The (k + 1)th order for eu can clearly be observed. However, the order of convergence of e¯u and eq may not be fully revealed from the tables. In order to closely examine the order of convergence, we plotted the numerical errors against mesh sizes in Figs. 6.1-6.3. The data sets are fitted by straight line in least-squares sense, and the slopes of the fitting lines are calculated, as shown atop the figures. Altogether, the slopes of fitting lines for e¯u is approximately (k + 2), and the slopes for eq is close to (k + 1). We would like to mention that, although the superconvergence rate is proved to be of order k + 3/2 in Section 4, we can observe the superconvergence of order k + 2 in our numerical experiments. The same phenomenon has been observed in [3].
6.2
Time history of the L2 error
In this subsection, we investigate the long time evolution of the L2 error of the LDG method. An energy conserving IPDG method was proposed in [17], which conserves a specifically 20
Table 6.1: Numerical errors and orders meshes and space P 1 . N eu e¯u 2 2 L error order L error 10 1.7099E-02 9.1769E-04 20 4.2397E-03 2.0119 8.8732E-08 40 1.0645E-03 1.9938 4.1486E-05 80 2.6556E-04 2.0031 3.8153E-06 160 6.6362E-05 2.0006 8.0124E-08
of LDG method for wave equation with uniform
Table 6.2: Numerical errors and orders meshes and space P 2 . N eu e¯u L2 error order L2 error 10 8.5041E-04 1.3099E-04 20 1.0541E-04 3.0121 3.4527E-06 40 1.3481E-05 2.9670 4.6501E-07 80 1.6795E-06 3.0048 2.3830E-08 160 2.0907E-07 3.0060 7.0408E-10
of LDG method for wave equation with uniform
eq order L error 4.2034E-02 13.336 8.1594E-03 -8.869 5.3769E-03 3.4428 1.0483E-03 5.5734 1.3333E-04 2
order 2.3650 0.6017 2.3587 2.9750
eq L2 error 4.2597E-03 3.7914E-04 6.4696E-05 7.2923E-06 7.2422E-07
order 5.2456 2.8924 4.2864 5.0809
order 3.4899 2.5510 3.1492 3.3319
Table 6.3: Numerical errors and orders of LDG method for wave equation with uniform meshes and space P 3 . N eu e¯u eq 2 2 2 L error order L error order L error order 10 3.2810e-05 5.4290e-06 2.0398e-04 20 2.0357e-06 4.0105 1.4383e-07 5.2382 9.8007e-06 4.3794 40 1.2881e-07 3.9822 6.5194e-09 4.4635 9.3343e-07 3.3923 80 8.0431e-09 4.0013 7.0231e-11 6.5365 2.6316e-08 5.1485
slope =3.1086
slope =2.076 −1
−2
−4
−1.5
−2.5
−5
−2
−3 −3.5 −4
−6 −7 −8
−4.5 −5 −2.5
log1 0 e q
−3
log1 0 e¯u
log1 0 e u
slope =2.0012 −1.5
−2
−1.5
log1 0 h
−1
−2.5 −3 −3.5
−9
−4
−10 −2.5
−4.5 −2.5
−2
−1.5
log1 0 h
−1
−2
−1.5
−1
log1 0 h
Figure 6.1: Numerical errors, eu , e¯u and eq , with P 1 space are plotted against mesh sizes. Errors are evaluated in L2 norm and marked by blue circles. Red line is the least square fitting of the data, with the slope of the fitting line displayed on the top of each panel. 21
slope =2.9985
slope =4.0712
−3
slope =3.0299
−3
−3.5
−2 −2.5
−4
−4
−3 −5 −3.5
−5.5 −6
−6
log1 0 e q
−5
log1 0 e¯u
log1 0 e u
−4.5
−7 −8
−6.5
−4 −4.5 −5 −5.5
−9 −7
−6 −10
−7.5 −8 −3
−2.5
−2
−1.5
−11 −3
−1
−6.5 −2.5
−2
−1.5
−7 −3
−1
−2.5
log1 0 h
log1 0 h
−2
−1.5
−1
log1 0 h
Figure 6.2: Numerical errors, eu , e¯u and eq , with P 2 space are plotted against mesh sizes. Errors are evaluated in L2 norm and marked by blue circles. Red line is the least square fitting of the data, with the slope of the fitting line displayed on the top of each panel.
slope =3.9952
slope =4.0038
slope =5.0502
−4
−5
−3.5
−4.5
−4 −6
−5
−4.5
−6 −6.5 −7
−5
log1 0 e q
−7
log1 0 e¯u
log1 0 e u
−5.5
−8
−9
−5.5 −6 −6.5
−7.5
−7 −10
−8 −8.5 −2
−7.5 −1.8
−1.6
−1.4
log1 0 h
−1.2
−1
−11 −2
−1.8
−1.6
−1.4
log1 0 h
−1.2
−1
−8 −2
−1.8
−1.6
−1.4
−1.2
−1
log1 0 h
Figure 6.3: Numerical errors, eu , e¯u and eq , with P 3 space are plotted against mesh sizes. Errors are evaluated in L2 norm and marked by blue circles. Red line is the least square fitting of the data, with the slope of the fitting line displayed on the top of each panel.
22
defined energy. Here we will compare the performance of these two methods. We consider again the wave equation utt = uxx ,
x ∈ [0, 2π]
with a periodic boundary condition u(0, t) = u(2π, t) for all t ≥ 0, and initial conditions u(x, 0) = esin x , ut (x, 0) = esin x cos x. This problem has the exact solution u = esin (x−t) . The LDG and IPDG methods are implemented with a uniform mesh with N cells, and the leap-frog time discretization, with ∆t = 0.6h2 . For the IPDG method, the interior penalty stabilization parameter is taken to be 40000/π. In order to examine the potential shape difference resulted from long time integration, both methods are run until T = 1000, with finite element spaces P 2 and P 3 , and N = 40, 80, respectively. In Fig. 6.4, the time evolution of L2 errors of both schemes are shown, with red color representing LDG method and blue color for IPDG method, and the errors are recorded every 10,000 iterations. The L2 errors of both schemes grow in a linear fashion, but the slope for IPDG method is much larger than that for LDG method, which almost stays as constant and is close to zero. In Fig. 6.4, the errors are plotted in log scale just for easy visualization. From the figure, one can see that for LDG method, the level of the errors are reduced by refining the mesh from N = 40 to N = 80, but the mesh refinement does not substantially reduce the errors of IPDG method due to the rapid growth. It can be observed from Fig. 6.4 that, up to T = 1000, the L2 error of IPDG method is greater than 10−1 , and this large error can easily be visualized by directly comparing the solutions of both methods. Fig. 6.5 displays the exact solution (red), the solution of LDG method (green) and the solution of IPDG method (blue) at T = 1000, for spaces P 2 and P 3 with N = 40. It can be seen that solution of LDG method overlaps with the exact solution, while the solution of IPDG method preserves the shape but has a phase shift. A finer mesh with N = 80 for IPDG was also tested, and the results look almost identical to Fig. 6.5, therefore not displayed here. We would like to remark that tuning the interior penalty stabilization parameter in IPDG method does result in solutions with different levels 23
P2, N=40
0
10
−1
−1
10
10
IPDG LDG
−3
10
−4
10
−3
10
−4
10
10
−5
0
IPDG LDG
−2
L2 error
L2 error
−2
10
10
P3, N=40
0
10
−5
200
400
600
800
10
1000
0
200
400
time 2
10
−1
−1
10
10
IPDG LDG
−2
L2 error
10
−3
10
−4
−3
10
−4
10
10
−5
−5
10
10
−6
0
IPDG LDG
−2
10
L2 error
1000
P , N=80
0
10
10
800
3
P , N=80
0
600
time
−6
200
400
600
800
10
1000
time
0
200
400
600
800
1000
time
Figure 6.4: Time history until T = 1000 of the L2 error of the numerical approximations obtained from the LDG and IPDG methods with k = 2, 3 and a uniform mesh with 40 and 80 cells. The L2 error on y-axis are presented in log scale. of phase shift. However, evident phase error similar to Fig. 6.5 can still be observed within a large range of the parameter.
7
Concluding remarks
In this paper, we developed and analyzed the LDG method for solving the wave equation. We have proved the optimal error estimates, superconvergence towards a particular projection of the exact solution, and its energy conserving property for the semi-discrete formulation. The leap-frog time discretization was used to obtain a fully discrete method, and the fully discrete method is shown to conserve discrete energy. Usually it is difficult to obtain DG schemes for wave equations which are non-dissipative and optimal high order accurate, without going to staggered mesh [4, 5] or using penalty with a parameter to adjust [17, 18]. Our scheme 24
T = 1000, P2, N=40
T = 1000, P3, N=40
3
3 IPDG LDG exact
2.5
2.5
2
u
u
2
1.5
1.5
1
1
0.5
0.5
0 0
IPDG LDG exact
1
2
3
4
5
6
0 0
7
x
1
2
3
4
5
6
7
x
Figure 6.5: Numerical approximations of the wave equation using LDG and IPDG methods. Comparison is made at T = 1000 with k = 2, 3 and N = 40. does not have parameters to tune and is defined on a single mesh. Numerical tests have demonstrated the optimal convergence rates and superconvergence which were proved in this paper. Numerical simulations with long time integration, in comparison with results from IPDG methods, confirmed that energy conservation property is important in order to preserve the phase and shape. Same techniques in the proofs of this paper can be generalized to variable coefficient problem and also to multi-dimensions, which constitute our current work in progress. Future work also includes designing a higher order time stepping method for the proposed LDG scheme, which conserves energy as the leap-frog discretization. Another direction to pursue is to consider the space-time DG method for the wave propagation problem.
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